Finite Group Databasebeta

← C31×#27037 #21483796

Subgroups — C31×#27037 #21483796

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2494C211yesmaximal, characteristic
2C31×#2360C211yesmaximal, characteristic
3C31×#2074C211yesmaximal, characteristic, complemented
4C31×#2066C211yesmaximal, characteristic, complemented
5C31×#2036C211yesmaximal, characteristic
6C31×#2035C211yesmaximal, characteristic
7C31×#1949C211yesmaximal, characteristic, complemented
8C31×#1941C211yesmaximal, characteristic, complemented
9C31×#1923C211yesmaximal, characteristic
10C31×#1921C211yesmaximal, characteristic, complemented
11C31×#1920C211yesmaximal, characteristic, complemented
12C31×#1865C211yesmaximal, characteristic, complemented
13C31×#1853C211yesmaximal, characteristic
14C31×#1823C211yesmaximal, characteristic
15C4×C31×#103C211yesmaximal, characteristic, complemented
16C2×C31×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C2×C31×#107C2211yescharacteristic
20C2×C31×#106C2211yescharacteristic
21C2×C31×#106C2211yescharacteristic
22C2×C31×#106C2211yescharacteristic
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×#103C2211yescharacteristic
25C2×C31×#103C2211yescharacteristic
26C2×C31×#103C2211yescharacteristic
27C2×C31×#103C2211yescharacteristic
28C2×C31×C22⋊C8C2211yescharacteristic
29C2×C31×C22⋊C8C2211yescharacteristic
30C2×C4×C8×C31C2211yescharacteristic
31C31×#401—21no—
32C31×#397—21no—
33C31×#391C2211yescharacteristic
34C31×#389—21no—
35C31×#389C2211yescharacteristic
36C31×#389C2211yescharacteristic
37C31×#389C2211yescharacteristic
38C31×#388—21no—
39C31×#387—21no—
40C31×#387—21no—
41C31×#385—21no—
42C31×#382C2211yescharacteristic
43C31×#380C2211yescharacteristic
44C31×#380C2211yescharacteristic
45C4×C4⋊C4×C31C2211yescharacteristic
46C4×C4⋊C4×C31C2211yescharacteristic
47C4×C22⋊C4×C31—21no—
48C4×C22⋊C4×C31C2211yescharacteristic
49C31×#340C2211yescharacteristic
50C31×#340C2211yescharacteristic
51C31×#340C2211yescharacteristic
52C31×#340C2211yescharacteristic
53C31×#340C2211yescharacteristic
54C31×#340C2211yescharacteristic
55C31×#340C2211yescharacteristic
56C31×#340C2211yescharacteristic
57C31×#336C2211yescharacteristic
58C31×#336C2211yescharacteristic
59C22×Q8×C31C2311yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31C2311yescharacteristic
62C22×C8×C31C2311yescharacteristic
63C22×C8×C31C2311yescharacteristic
64C22×C8×C31C2311yescharacteristic
65C22×C8×C31C2311yescharacteristic
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C4×Q8×C31—22no—
69C4×Q8×C31—41no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—41no—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31D411yescharacteristic
90C2×C42×C31—21no—
91C2×C42×C31—21no—
92C2×C42×C31—21no—
93C2×C42×C31C2311yescharacteristic
94C2×C42×C31C2311yescharacteristic
95C31×#107—22no—
96C31×#106—22no—
97C31×#106—22no—
98C31×#106—41no—
99C31×C4⋊C8—22no—
100C31×#103—22no—
101C31×#103—22no—
102C31×#103—22no—
103C31×#103—41no—
104C31×C22⋊C8—22no—
105C31×C22⋊C8—41no—
106C4×C8×C31—41no—
107C31×#95—21no—
108C31×#95—21no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95—21no—
118C31×#95D411yescharacteristic
119C31×#95D411yescharacteristic
120C24×C31C2×D411yescharacteristic
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—21no—
126C2×Q8×C31#4112yes—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—42no—
142C22×C4×C31—42no—
143C22×C4×C31—81no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2411yescharacteristic
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—21no—
158C2×C8×C31—21no—
159C2×C8×C31—21no—
160C2×C8×C31—21no—
161C2×C8×C31#4112yes—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—42no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31—42no—
178C4⋊C4×C31—81no—
179C4⋊C4×C31—81no—
180C4⋊C4×C31#4112yes—
181C4⋊C4×C31#4112yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—42no—
186C22⋊C4×C31—81no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—21no—
191C42×C31—42no—
192C42×C31—81no—
193C42×C31—81no—
194#27037C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31#13512yes—
202C23×C31C22×D411yescharacteristic
203Q8×C31—22no—
204Q8×C31—22no—
205Q8×C31—41no—
206Q8×C31—41no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—84no—
250C2×C4×C31#13712yes—
251C2×C4×C31C2×#4111yescharacteristic
252C2×C4×C31C2×#4111yescharacteristic
253C2×C4×C31C2×#4111yescharacteristic
254C2×C4×C31C2×#4111yescharacteristic
255C2×C4×C31H(5,𝔽2)11yescharacteristic
256C2×C4×C312−1+411yescharacteristic
257C8×C31—22no—
258C8×C31—41no—
259C8×C31—41no—
260C8×C31—41no—
261#2494C2×C3111yescharacteristic
262#2360C2×C3111yescharacteristic
263#2074C2×C3111yescharacteristic, complemented
264#2066C2×C3111yescharacteristic, complemented
265#2036C2×C3111yescharacteristic
266#2035C2×C3111yescharacteristic
267#1949C2×C3111yescharacteristic, complemented
268#1941C2×C3111yescharacteristic, complemented
269#1923C2×C3111yescharacteristic
270#1921C2×C3111yescharacteristic, complemented
271#1920C2×C3111yescharacteristic, complemented
272#1865C2×C3111yescharacteristic, complemented
273#1853C2×C3111yescharacteristic
274#1823C2×C3111yescharacteristic
275C4×#103C2×C3111yescharacteristic, complemented
276C22×C31—22no—
277C22×C31—22no—
278C22×C31—22no—
279C22×C31—41no—
280C22×C31—41no—
281C22×C31—41no—
282C22×C31—41no—
283C22×C31#52511yescharacteristic
284C22×C31#54611yescharacteristic
285C22×C31#54711yescharacteristic
286C22×C31C2×#13511yescharacteristic
287C22×C31C2×#13711yescharacteristic
288C22×C31#57711yescharacteristic
289C22×C31#57711yescharacteristic
290C4×C31—22no—
291C4×C31—41no—
292C4×C31—41no—
293C4×C31—42no—
294C4×C31—41no—
295C4×C31—42no—
296C4×C31—41no—
297C4×C31—42no—
298C4×C31—81no—
299C4×C31—81no—
300C4×C31—81no—
301C4×C31—81no—
302C4×C31—81no—
303C4×C31#52911yescharacteristic
304C4×C31#53911yescharacteristic
305C4×C31#55311yescharacteristic
306C4×C31#55611yescharacteristic
307C2×C22⋊Q8C22×C3111yescharacteristic
308C2×C4×Q8C22×C3111yescharacteristic
309C22×C4⋊C4C22×C3111yescharacteristic
310C2×#107C22×C3111yescharacteristic
311C2×#106C22×C3111yescharacteristic
312C2×#106C22×C3111yescharacteristic
313C2×#106C22×C3111yescharacteristic
314C2×C4⋊C8C22×C3111yescharacteristic
315C2×#103C22×C3111yescharacteristic
316C2×#103C22×C3111yescharacteristic
317C2×#103C22×C3111yescharacteristic
318C2×#103C22×C3111yescharacteristic
319C2×C22⋊C8C22×C3111yescharacteristic
320C2×C22⋊C8C22×C3111yescharacteristic
321C2×C4×C8C22×C3111yescharacteristic
322#401—21no—
323#397—21no—
324#391C22×C3111yescharacteristic
325#389—21no—
326#389C22×C3111yescharacteristic
327#389C22×C3111yescharacteristic
328#389C22×C3111yescharacteristic
329#388—21no—
330#387—21no—
331#387—21no—
332#385—21no—
333#382C22×C3111yescharacteristic
334#380C22×C3111yescharacteristic
335#380C22×C3111yescharacteristic
336C4×C4⋊C4C22×C3111yescharacteristic
337C4×C4⋊C4C22×C3111yescharacteristic
338C4×C22⋊C4—21no—
339C4×C22⋊C4C22×C3111yescharacteristic
340#340C22×C3111yescharacteristic
341#340C22×C3111yescharacteristic
342#340C22×C3111yescharacteristic
343#340C22×C3111yescharacteristic
344#340C22×C3111yescharacteristic
345#340C22×C3111yescharacteristic
346#340C22×C3111yescharacteristic
347#340C22×C3111yescharacteristic
348#336C22×C3111yescharacteristic
349#336C22×C3111yescharacteristic
350C2×C31—42no—
351C2×C31#246611yescharacteristic
352C2×C31#310811yescharacteristic
353C2×C31#312511yescharacteristic
354C2×C31#329611yescharacteristic
355C2×C31#332011yescharacteristic
356C2×C31#333711yescharacteristic
357C2×C31#335611yescharacteristic
358C22×Q8C23×C3111yescharacteristic
359C23×C4—21no—
360C23×C4C23×C3111yescharacteristic
361C22×C8C23×C3111yescharacteristic
362C22×C8C23×C3111yescharacteristic
363C22×C8C23×C3111yescharacteristic
364C22×C8C23×C3111yescharacteristic
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C4×Q8—22no—
368C4×Q8—41no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—41no—
377C2×C4⋊C4C23×C3111yescharacteristic
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C4⋊C4C23×C3111yescharacteristic
380C2×C4⋊C4C23×C3111yescharacteristic
381C2×C4⋊C4C23×C3111yescharacteristic
382C2×C4⋊C4C23×C3111yescharacteristic
383C2×C4⋊C4C23×C3111yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4D4×C3111yescharacteristic
388C2×C22⋊C4D4×C3111yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42C23×C3111yescharacteristic
393C2×C42C23×C3111yescharacteristic
394#107—22no—
395#106—22no—
396#106—22no—
397#106—41no—
398C4⋊C8—22no—
399#103—22no—
400#103—22no—
401#103—22no—
402#103—41no—
403C22⋊C8—22no—
404C22⋊C8—41no—
405C4×C8—41no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95D4×C3111yescharacteristic
418#95D4×C3111yescharacteristic
419C31#2703711yescharacteristic, direct product factor, Sylow
420C24C2×D4×C3111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#41×C3112yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C24×C3111yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8#41×C3112yes—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4#41×C3112yes—
481C4⋊C4#41×C3112yes—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—81no—
493C42—81no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C31×#13512yes—
501C23C22×D4×C3111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4C31×#13712yes—
550C2×C4C2×#41×C3111yescharacteristic
551C2×C4C2×#41×C3111yescharacteristic
552C2×C4C2×#41×C3111yescharacteristic
553C2×C4C2×#41×C3111yescharacteristic
554C2×C4C31×H(5,𝔽2)11yescharacteristic
555C2×C4C31×2−1+411yescharacteristic
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C31×#52511yescharacteristic
568C22C31×#54611yescharacteristic
569C22C31×#54711yescharacteristic
570C22C2×C31×#13511yescharacteristic
571C22C2×C31×#13711yescharacteristic
572C22C31×#57711yescharacteristic
573C22C31×#57711yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4C31×#52911yescharacteristic
588C4C31×#53911yescharacteristic
589C4C31×#55311yescharacteristic
590C4C31×#55611yescharacteristic
591C2—42no—
592C2C31×#246611yescharacteristic
593C2C31×#310811yescharacteristic
594C2C31×#312511yescharacteristic
595C2C31×#329611yescharacteristic
596C2C31×#332011yescharacteristic
597C2C31×#333711yescharacteristic
598C2C31×#335611yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C22⋊C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C31×#2074C2semidirect
4. C31×#2066C2semidirect
7. C31×#1949C2semidirect
8. C31×#1941C2semidirect
10. C31×#1921C2semidirect
11. C31×#1920C2semidirect
12. C31×#1865C2semidirect
15. C4×C31×#103C2semidirect
194. #27037C31direct (N × H)
263. #2074C2×C31semidirect
264. #2066C2×C31semidirect
267. #1949C2×C31semidirect
268. #1941C2×C31semidirect
270. #1921C2×C31semidirect
271. #1920C2×C31semidirect
272. #1865C2×C31semidirect
275. C4×#103C2×C31semidirect
419. C31#27037direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#24942
C31×#23602
C31×#20742
C31×#20662
C31×#20362
C31×#20352
C31×#19492
C31×#19412
C31×#19232
C31×#19212
C31×#19202
C31×#18652
C31×#18532
C31×#18232
C4×C31×#1032
#2703731